C&SA | Contents

Volume 62 >>> № 1 JANUARY — FEBRUARY 2026

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UDC 519.2

M.I. Schlesinger
Institute for Information Technologies and Systems of the NAS of Ukraine,
Kyiv, Ukraine, schles@irtc.org.ua

E.V. Vodolazskiy
Institute for Information Technologies and Systems of the NAS of Ukraine,
Kyiv, Ukraine, waterlaz@gmail.com

ASYMMETRICAL GENERALISATION OF CHEBYSHEV’S INEQUALITY

Abstract. The paper explores a family of generalizations of Chebyshev’s inequality, focusing on asymmetric cases where the deviation region of interest is not symmetric about the mean. The well-known results of Cantelli and Selberg are revisited as natural extensions of Chebyshev’s classical bound. A unified linear programming framework is proposed that provides a geometric intuition and derivation of these inequalities. This leads to concise derivations of the inequalities. These results highlight the versatility and power of linear programming in deriving probabilistic bounds under various constraints.

Keywords: Chebyshev’s inequality, Cantelli’s inequality, Selberg’s inequality, probability bounds.

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